Physics-Informed Linear Models (PILM)

• Physics-Informed Linear Model (PILM) is a framework that combines linear predictors and fixed basis functions with physical laws to enforce PDE, ODE, and boundary conditions.
• It yields analytical least-squares solutions by minimizing quadratic data and physics residuals and can be interpreted via kernel ridge regression in an RKHS setting.
• PILM demonstrates practical benefits in regression, inverse problems, and control, with applications ranging from crustal strain-rate estimation to feature-engineered physical modeling.

Physics-Informed Linear Model (PILM) denotes a class of physics-informed machine learning methods in which the predictor is linear in coefficients, parameters, or feature-space coordinates, while physical prior knowledge is encoded through partial differential equations, ordinary differential equations, boundary or initial conditions, conservation laws, symmetry constraints, or physically constructed feature maps. In the narrow formulation explicitly titled “Physics-Informed Linear Model (PILM): Analytical Representations and Application to Crustal Strain Rate Estimation,” the unknown field is represented as a linear combination of fixed basis functions and estimated by minimizing quadratic residuals of data and physics, yielding an analytical least-squares solution (Okazaki, 16 Jul 2025). In a broader functional-analytic formulation, physics-informed regularization over Sobolev spaces induces a reproducing kernel Hilbert space (RKHS), so the estimator becomes kernel ridge regression with a PDE-dependent kernel and remains linear in its finite coefficient vector (Doumèche et al., 2024).

1. Conceptual position within physics-informed machine learning

Physics-informed machine learning (PIML) is defined as a learning paradigm that leverages empirical data together with prior physical knowledge to improve performance on tasks involving physical mechanisms. A standard supervised formulation augments data misfit with a physics penalty or constraint,

$$\min_{f \in \mathcal{H}_p} \Big( \mathcal{L}_{\text{data}}(f;\mathcal{D}) + \lambda_{\text{phys}}\mathcal{L}_{\text{physics}}(f) + \Omega(f) \Big),$$

where the physical term may encode PDE or ODE residuals, boundary or initial conditions, conservation laws, or symmetry constraints (Hao et al., 2022).

Within that framework, a PILM is the specialization in which the hypothesis class is linear: either directly linear in parameters, as in

$f_\theta(x)=\theta^\top \phi(x),$

or linear in a function-space representation such as an RKHS expansion. The surveys do not use the exact term “Physics-Informed Linear Model,” but they describe the precise niche it occupies: a physics-informed supervised method with a linear hypothesis space and physics-based regularization or constraints, suitable for regression, interpolation, inverse problems, and control (Meng et al., 2022).

A related distinction is between penalty-based physics-informed learning and architecture-based physics augmentation. If a physical property is discriminative, meaning there exists an efficiently computable operator $\hat{L}$ such that $P(f)$ is true iff $\hat{L}f=0$, then a PILM can enforce physics through a residual penalty. If the property is generative rather than discriminative, a plausible implication is that a pure residual-based PILM may be insufficient, and a decomposition into a physics-generated component and a residual component may be more natural (Liu et al., 2021).

2. Core mathematical formulations

In the analytical basis-expansion formulation, the unknown field is written as

$$u(x;\mathbf{c})=\sum_{j=1}^M c_j\,\phi_j(x)=\boldsymbol{\Phi}(x)^T\mathbf{c},$$

with fixed basis functions $\phi_j$. For a linear PDE

$\mathcal{A}[u](x)=f(x), \qquad x\in\Omega,$

and boundary condition

$\mathcal{B}[u](x)=g(x), \qquad x\in\partial\Omega,$

all residuals are linear in $\mathbf{c}$. The total quadratic loss can therefore be written as

$f_\theta(x)=\theta^\top \phi(x),$0

where $f_\theta(x)=\theta^\top \phi(x),$1 collects linear observation, boundary, or initial-condition operators and $f_\theta(x)=\theta^\top \phi(x),$2 collects PDE or regularization contributions. The global minimizer satisfies

$f_\theta(x)=\theta^\top \phi(x),$3

which is the characteristic closed-form PILM solution (Okazaki, 16 Jul 2025).

The same structure appears in the nonparametric regression formulation

$f_\theta(x)=\theta^\top \phi(x),$4

with $f_\theta(x)=\theta^\top \phi(x),$5, $f_\theta(x)=\theta^\top \phi(x),$6, and a known linear differential operator

$f_\theta(x)=\theta^\top \phi(x),$7

encoding the physical prior $f_\theta(x)=\theta^\top \phi(x),$8. The estimator minimizes

$f_\theta(x)=\theta^\top \phi(x),$9

or equivalently

$\hat{L}$0

Here the physical inconsistency of a candidate $\hat{L}$1 is measured by $\hat{L}$2, and $\hat{L}$3 plays the role of modeling error when the PDE is not exact (Doumèche et al., 2024).

These two presentations are mathematically consistent. In both, linearity of the operator $\hat{L}$4 or $\hat{L}$5 makes the physics term quadratic in the unknown coefficients. This is the defining algebraic property that distinguishes PILM from nonlinear PINN training.

3. RKHS, kernel, and operator-theoretic interpretation

For linear differential priors, the physics-informed regularizer defines an RKHS norm. There exists a positive, self-adjoint, compact operator

$\hat{L}$6

such that

$\hat{L}$7

With the induced inner product, $\hat{L}$8 becomes an RKHS whose kernel is

$\hat{L}$9

where $P(f)$0 are eigenpairs of $P(f)$1. The regularized empirical risk then becomes exactly kernel ridge regression in that RKHS (Doumèche et al., 2024).

The representer theorem yields the finite-dimensional form

$P(f)$2

so the estimator is linear in the coefficient vector $P(f)$3. In this sense, the PDE-informed kernel method is itself a Physics-Informed Linear Model: the basis functions are the kernel sections $P(f)$4, and the physical law is encoded in the kernel rather than in an explicit neural architecture. The kernel admits a weak PDE characterization: for each fixed $P(f)$5, $P(f)$6 is the unique solution of a weak problem combining Sobolev and $P(f)$7-terms, so the kernel itself is PDE-induced (Doumèche et al., 2024).

The dissertation “Physics-informed machine learning: A mathematical framework with applications to time series forecasting” extends the same viewpoint. It writes the physics-informed risk as

$P(f)$8

and interprets the resulting estimator as kernel ridge regression with a physics-informed kernel. It also introduces the Physics-Informed Kernel Learner (PIKL), which approximates the exact kernel by low-frequency Fourier modes

$P(f)$9

thereby reducing the problem to finite-dimensional linear regression with quadratic regularization in the coefficients $\hat{L}f=0$0 (Doumèche, 11 Jul 2025).

4. Representative PILM constructions

A first major construction is the analytical basis-expansion PILM. The 2025 PILM paper uses fixed basis functions, especially cubic B-splines and tensor-product cubic B-splines, so that derivatives and integrals of residuals can be computed analytically rather than by automatic differentiation and collocation. It treats forward problems, inverse problems, underdetermined systems, and cases with uncertain boundary conditions. Illustrative examples include a damped oscillation ODE,

$\hat{L}f=0$1

and a 1D diffusion PDE,

$\hat{L}f=0$2

In the geodetic application, horizontal velocity fields are expanded on a regular $\hat{L}f=0$3 grid with tensor-product cubic B-splines, using 529 basis functions per component and 1058 model parameters in total (Okazaki, 16 Jul 2025).

A second construction is the feature-space PILM. “Physics-informed features in supervised machine learning” uses a nonlinear feature map $\hat{L}f=0$4 built from dimensional analysis and known physical laws, followed by linear ridge regression,

$\hat{L}f=0$5

This preserves linearity in $\hat{L}f=0$6 while moving the physics into feature engineering. Examples include Bernoulli-inspired pressure features

$\hat{L}f=0$7

binary-system binding-energy features,

$\hat{L}f=0$8

and solar-flare forecasting features such as $\hat{L}f=0$9. In the Bernoulli example, feature ranking isolates the physically relevant terms and de-standardized coefficients recover values close to $$u(x;\mathbf{c})=\sum_{j=1}^M c_j\,\phi_j(x)=\boldsymbol{\Phi}(x)^T\mathbf{c},$$0, $$u(x;\mathbf{c})=\sum_{j=1}^M c_j\,\phi_j(x)=\boldsymbol{\Phi}(x)^T\mathbf{c},$$1, and $$u(x;\mathbf{c})=\sum_{j=1}^M c_j\,\phi_j(x)=\boldsymbol{\Phi}(x)^T\mathbf{c},$$2 for the pressure, dynamic-pressure, and hydrostatic terms (Lampani et al., 23 Apr 2025).

A third construction arises in dynamical systems and control. There the PILM is a linear state-space model

$$u(x;\mathbf{c})=\sum_{j=1}^M c_j\,\phi_j(x)=\boldsymbol{\Phi}(x)^T\mathbf{c},$$3

whose matrices are learned from data while satisfying structure, conservation, stability, or invariance constraints. Physics enters through hard constraints, quadratic penalties, or matrix factorizations that bound singular values or spectral radii. This places PILM directly inside model predictive control, predictive safety filters, robust MPC, and digital-twin pipelines (Nghiem et al., 2023).

A closely related neighboring methodology is physics-informed neural modeling of linear PDEs. In dynamic linear elasticity, the governing equations remain those of linear continuum elasticity, with outputs including displacement and stress and losses enforcing momentum balance and Hooke’s law. That line does not present itself as a PILM in the strict basis-expansion sense, but it shows that once the governing operator is linear, physics-informed formulations with linear field equations admit the same residual structure and many of the same implementation choices (Kag et al., 2023).

5. Statistical behavior, regularization, and empirical findings

The kernel-theoretic analysis establishes that physics-informed linear regularization is not merely heuristic. Under suitable assumptions, the minimizer of the regularized risk converges at least at the Sobolev minimax rate. In particular, if

$$u(x;\mathbf{c})=\sum_{j=1}^M c_j\,\phi_j(x)=\boldsymbol{\Phi}(x)^T\mathbf{c},$$4

then

$$u(x;\mathbf{c})=\sum_{j=1}^M c_j\,\phi_j(x)=\boldsymbol{\Phi}(x)^T\mathbf{c},$$5

More generally, the upper bound contains a bias term involving $$u(x;\mathbf{c})=\sum_{j=1}^M c_j\,\phi_j(x)=\boldsymbol{\Phi}(x)^T\mathbf{c},$$6 and a variance term involving the effective dimension $$u(x;\mathbf{c})=\sum_{j=1}^M c_j\,\phi_j(x)=\boldsymbol{\Phi}(x)^T\mathbf{c},$$7. The explicit implication drawn in the paper is that faster rates than the Sobolev minimax rate can occur when the physical error is small and the PDE prior improves eigenvalue decay (Doumèche et al., 2024).

That speed-up is explicit in the one-dimensional example with $$u(x;\mathbf{c})=\sum_{j=1}^M c_j\,\phi_j(x)=\boldsymbol{\Phi}(x)^T\mathbf{c},$$8, $$u(x;\mathbf{c})=\sum_{j=1}^M c_j\,\phi_j(x)=\boldsymbol{\Phi}(x)^T\mathbf{c},$$9, $\phi_j$0, and $\phi_j$1. With

$\phi_j$2

the estimator obeys

$\phi_j$3

When $\phi_j$4, the first term disappears and the rate becomes essentially parametric $\phi_j$5 up to logarithmic factors. The numerical illustration reports slopes $\phi_j$6 for perfect modeling and $\phi_j$7 for imperfect modeling on log-log plots, matching the theoretical distinction between exact and approximate physics (Doumèche et al., 2024).

Feature-space PILMs also report empirical gains. For Bernoulli regression, the PIF-based model yields MAE $\phi_j$8 versus $\phi_j$9 and MSE $\mathcal{A}[u](x)=f(x), \qquad x\in\Omega,$0 versus $\mathcal{A}[u](x)=f(x), \qquad x\in\Omega,$1 at $\mathcal{A}[u](x)=f(x), \qquad x\in\Omega,$2 noise. For binary-system classification, the SPIF model reports accuracy $\mathcal{A}[u](x)=f(x), \qquad x\in\Omega,$3 versus $\mathcal{A}[u](x)=f(x), \qquad x\in\Omega,$4, HSS $\mathcal{A}[u](x)=f(x), \qquad x\in\Omega,$5 versus $\mathcal{A}[u](x)=f(x), \qquad x\in\Omega,$6, and TSS $\mathcal{A}[u](x)=f(x), \qquad x\in\Omega,$7 versus $\mathcal{A}[u](x)=f(x), \qquad x\in\Omega,$8; for solar-flare forecasting, the reported comparison is accuracy $\mathcal{A}[u](x)=f(x), \qquad x\in\Omega,$9 versus $\mathcal{B}[u](x)=g(x), \qquad x\in\partial\Omega,$0, TSS $\mathcal{B}[u](x)=g(x), \qquad x\in\partial\Omega,$1 versus $\mathcal{B}[u](x)=g(x), \qquad x\in\partial\Omega,$2, and HSS $\mathcal{B}[u](x)=g(x), \qquad x\in\partial\Omega,$3 versus $\mathcal{B}[u](x)=g(x), \qquad x\in\partial\Omega,$4 (Lampani et al., 23 Apr 2025).

A central empirical caution emerges from crustal strain-rate estimation. In that application, mathematical regularization based on smoothness,

$\mathcal{B}[u](x)=g(x), \qquad x\in\partial\Omega,$5

outperforms physical regularization based on elastic equilibrium under the Bayesian marginal-likelihood criterion. The paper reports that mathematical regularization yields higher log marginal likelihood, smaller inferred noise standard deviation, and smaller data misfit; physical regularization produces blurred strain-rate features and artificially large strain rates at domain margins. This is a direct reminder that “physics-informed” does not automatically imply superior regularization if the chosen physical surrogate is misaligned with the inference target (Okazaki, 16 Jul 2025).

6. Scope, limitations, and open problems

The strongest current theoretical results require linear differential operators. In the kernel formulation, linearity of $\mathcal{B}[u](x)=g(x), \qquad x\in\partial\Omega,$6 is crucial because it underpins weak PDE theory, the existence of the operator $\mathcal{B}[u](x)=g(x), \qquad x\in\partial\Omega,$7, and the RKHS reduction. Nonlinear PDEs are not covered by that machinery. The analytical basis-expansion PILM has a parallel restriction: it is analytically solvable precisely because the model is linear in coefficients, the operators are linear, and the loss is quadratic (Doumèche et al., 2024).

The analytical PILM of 2025 is further limited to linear PDE or ODE systems with constant coefficients, low-dimensional rectangular domains, and fixed basis functions such as cubic B-splines. Its computational profile is governed by matrix construction and inversion, with building costs roughly $\mathcal{B}[u](x)=g(x), \qquad x\in\partial\Omega,$8 and linear-system solution $\mathcal{B}[u](x)=g(x), \qquad x\in\partial\Omega,$9. The paper also identifies a representation-capacity limitation: when the basis is too coarse, systematic approximation error remains even though the optimization is exact (Okazaki, 16 Jul 2025).

In the kernel setting, the framework allows any dimension $\mathbf{c}$0 and Lipschitz domain in principle, but explicit kernel computation and eigenvalue bounds become hard in high dimension or on complex domains. The dissertation and the kernel paper therefore emphasize approximations such as low-frequency Fourier expansions and GPU-oriented linear algebra, together with open questions on numerical kernel construction, source conditions for $\mathbf{c}$1, and extensions to nonlinear PDEs (Doumèche, 11 Jul 2025).

A recurring misconception is that PILM is simply a small PINN. The literature suggests a sharper distinction. PILM is inherently a linear model in coefficients or feature-space coordinates; PINNs use nonlinear parameterizations even when the underlying PDE is linear. Another misconception is that all useful physical priors are naturally expressible as residual penalties. The discriminative-versus-generative distinction shows why that is not always the case: some properties admit an efficient operator $\mathbf{c}$2 and fit naturally into residual-based PILM, whereas others are more naturally enforced by structural decomposition into a physics-generated component and a residual component (Liu et al., 2021).

The main development directions already identified in the literature are numerical strategies to compute PDE-induced kernels, extension to nonlinear PDEs or polynomial operators, hybrid linear–nonlinear models in which a PILM backbone is augmented by a nonlinear residual, structured block-partitioned linear models for multi-physics systems, and Bayesian or Gaussian-process constructions whose covariance is itself physics-informed (Nghiem et al., 2023). A plausible implication is that the enduring role of PILM will be as a mathematically transparent regime within PIML: one in which interpretability, convexity, closed-form estimation, and statistical analysis remain accessible without abandoning explicit physical structure.